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In this paper, we introduce and investigate the concepts of down continuity and down compactness. A real valued function $f$ on a subset $E$ of $\R$, the set of real numbers is down continuous if it preserves downward half Cauchy sequences, i.e. the sequence $(f(\alpha_{n}))$ is downward half Cauchy whenever $(\alpha_{n})$ is a downward half Cauchy sequence of points in $E$, where a sequence $(\alpha_{ k})$ of points in $\R$ is called downward half Cauchy if for every $\varepsilon>0$ there exists an $n_{0}\in{\N}$ such that $\alpha_{m}-\alpha_{n} <\varepsilon$ for $m \geq n \geq n_0$. It turns out that the set of down continuous functions is a proper subset of the set of continuous functions., Comment: 10 pages

A real valued function defined on a subset $E$ of $\mathbb{R}$, the set of real numbers, is $\rho$-statistically downward continuous if it preserves $\rho$-statistical downward quasi-Cauchy sequences of points in $E$, where a sequence $(\alpha_{k})$ of real numbers is called ${\rho}$-statistically downward quasi-Cauchy if $\lim_{n\rightarrow\infty}\frac{1}{\rho_{n} }|\{k\leq n: \Delta \alpha_{k} \geq \varepsilon\}|=0 $ for every $\varepsilon>0$, in which $(\rho_{n})$ is a non-decreasing sequence of positive real numbers tending to $\infty$ such that $\limsup _{n} \frac{\rho_{n}}{n}<\infty $, $\Delta \rho_{n}=O(1)$, and $\Delta \alpha _{k} =\alpha _{k+1} - \alpha _{k}$ for each positive integer $k$. It turns out that a function is uniformly continuous if it is $\rho$-statistical downward continuous on an above bounded set., Comment: 16 pages. arXiv admin note: text overlap with arXiv:1710.00515

In this paper, we introduce and investigate a concept of Abel statistical continuity. A real valued function $f$ is Abel statistically continuous on a subset $E$ of $\R$, the set of real numbers, if it preserves Abel statistical convergent sequences, i.e. $(f(p_{k}))$ is Abel statistically convergent whenever $(p_{k})$ is an Abel statistical convergent sequence of points in $E$, where a sequence $(p_{k})$ of point in $\R$ is called Abel statistically convergent to a real number $L$ if Abel density of the set $\{k\in{\N}: |p_{k}-L|\geq\varepsilon \}$ is $0$ for every $\varepsilon>0$. Some other types of continuities are also studied and interesting results are obtained., Comment: 11 pages. arXiv admin note: text overlap with arXiv:1101.1440

In this paper, we introduce a concept of statistically $p$-quasi-Cauchyness of a real sequence in the sense that a sequence $(\alpha_{k})$ is statistically $p$-quasi-Cauchy if $\lim_{n\rightarrow\infty}\frac{1}{n}|\{k\leq n: |\alpha_{k+p}-\alpha_{k}|\geq{\varepsilon}\}|=0$ for each $\varepsilon>0$. A function $f$ is called statistically $p$-ward continuous on a subset $A$ of the set of real umbers $\mathbb{R}$ if it preserves statistically $p$-quasi-Cauchy sequences, i.e. the sequence $f(\textbf{x})=(f(\alpha_{n}))$ is statistically $p$-quasi-Cauchy whenever $\boldsymbol\alpha=(\alpha_{n})$ is a statistically $p$-quasi-Cauchy sequence of points in $A$. It turns out that a real valued function $f$ is uniformly continuous on a bounded subset $A$ of $\mathbb{R}$ if there exists a positive integer $p$ such that $f$ preserves statistically $p$-quasi-Cauchy sequences of points in $A$., Comment: 13 pages

In this paper, the concept of an $N_{\theta}^{2}$ quasi-Cauchy sequence is introduced. We proved interesting theorems related to $N_{\theta}^{2}$-quasi-Cauchy sequences. A real valued function $f$ defined on a subset $A$ of $\mathbb{R}$, the set of real numbers, is $N_{\theta}^{2}$ ward continuous on $A$ if it preserves $N_{\theta}^{2}$ quasi-Cauchy sequences of points in $A$, i.e. $(f( \alpha_{k}))$ is an $N_{\theta}^{2}$ quasi-Cauchy sequence whenever $(\alpha_{k})$ is an $N_{\theta}^{2}$ quasi-Cauchy sequences of points in $A$, where a sequence $(\alpha_{k})$ is called $N_{\theta}^{2}$ quasi-Cauchy if $(\Delta^{2} \alpha_{k})$ is an $N_{\theta}$ quasi-Cauchy sequence where $\Delta^{2}\alpha_{k}=\alpha_{k+2}-2\alpha_{k+1}+\alpha_{k}$ for each positive integer $k$., Comment: 11 pages

A function $f$ defined on a subset $E$ of a two normed space $X$ is statistically ward continuous if it preserves statistically quasi-Cauchy sequences of points in $E$ where a sequence $(x_n)$ is statistically quasi-Cauchy if $(\Delta x_{n})$ is a statistically null sequence. A subset $E$ of $X$ is statistically ward compact if any sequence of points in $E$ has a statistically quasi-Cauchy subsequence. In this paper, new kinds of continuities are investigated in two normed spaces. It turns out that uniform limit of statistically ward continuous functions is again statistically ward continuous., Comment: 11 pages

A double sequence $\textbf{x}=\{x_{k,l}\}$ of points in $\textbf{R}$ is slowly oscillating if for any given $\varepsilon>0$, there exist $\alpha=\alpha(\varepsilon)>0$, $\delta=\delta (\varepsilon) >0$, and $N=N(\varepsilon)$ such that $|x_{k,l}-x_{s,t}|<\varepsilon$ whenever $k,l\geq N(\varepsilon)$ and $k\leq s \leq (1+\alpha)k$, $l\leq t \leq (1+\delta)l$. We study continuity type properties of factorable double functions defined on a double subset $A\times A$ of $\textbf{R}^{2}$ into $\textbf{R}$, and obtain interesting results related to uniform continuity, sequential continuity, and a newly introduced type of continuity of factorable double functions defined on a double subset $A\times A$ of $\textbf{R}^{2}$ into $\textbf{R}$., Comment: 11 pages. arXiv admin note: substantial text overlap with arXiv:1312.6602

A double sequence $\{x_{k,l}\}$ is quasi-Cauchy if given an $\epsilon > 0$ there exists an $N \in {\bf N}$ such that $$\max_{r,s= 1\mbox{ and/or} 0} \left \{|x_{k,l} - x_{k+r,l+s}|< \epsilon\right \} .$$ We study continuity type properties of factorable double functions defined on a double subset $A\times A$ of ${\bf R}^{2}$ into $\textbf{R}$, and obtain interesting results related to uniform continuity, sequential continuity, continuity, and a newly introduced type of continuity of factorable double functions defined on a double subset $A\times A$ of ${\bf R}^{2}$ into $\textbf{R}$., Comment: 11 pages

A real valued function $f$ defined on a subset $E$ of $\textbf{R}$, the set of real numbers, is statistically upward continuous if it preserves statistically upward half quasi-Cauchy sequences, is statistically downward continuous if it preserves statistically downward half quasi-Cauchy sequences; and a subset $E$ of $\textbf{R}$, is statistically upward compact if any sequence of points in $E$ has a statistically upward half quasi-Cauchy subsequence, is statistically downward compact if any sequence of points in $E$ has a statistically downward half quasi-Cauchy subsequence where a sequence $(x_{n})$ of points in $\textbf{R}$ is called statistically upward half quasi-Cauchy if \[ \lim_{n\rightarrow\infty}\frac{1}{n}|\{k\leq n: x_{k}-x_{k+1}\geq \varepsilon\}|=0 \] is statistically downward half quasi-Cauchy if \[ \lim_{n\rightarrow\infty}\frac{1}{n}|\{k\leq n: x_{k+1}-x_{k}\geq \varepsilon\}|=0 \] for every $\varepsilon>0$. We investigate statistically upward continuity, statistically downward continuity, statistically upward half compactness, statistically downward half compactness and prove interesting theorems. It turns out that uniform limit of a sequence of statistically upward continuous functions is statistically upward continuous, and uniform limit of a sequence of statistically downward continuous functions is statistically downward continuous., Comment: 25 pages. arXiv admin note: substantial text overlap with arXiv:1205.3674, arXiv:1103.1230, arXiv:1102.1531, arXiv:1305.0697

A function $f$ defined on a 2-normed space $ (X,||.,.||)$ is ward continuous if it preserves quasi-Cauchy sequences where a sequence $(x_n)$ of points in $X$ is called quasi-Cauchy if $lim_{n\rightarrow\infty}||\Delta x_{n},z||=0$ for every $z\in X$. Some other kinds of continuties are also introduced via quasi-Cauchy sequences in 2-normed spaces. It turns out that uniform limit of ward continuous functions is again ward continuous., Comment: 15 pages

In this paper, we prove that any ideal ward continuous function is uniformly continuous either on an interval or on an ideal ward compact subset of $\textbf{R}$. A characterization of uniform continuity is also given via ideal quasi-Cauchy sequences., Comment: This paper has been withdrawn by the author due to the publication with the title "A variation on ward continuity" in the peer reviewed refereed journal Filomat Filomat 27:8 (2013), 1545-1549 at the following link http://www.pmf.ni.ac.rs/pmf/publikacije/filomat/2013/27-8/F27-8-17.pdf with DOI 10.2298/FIL1308545C

The main object of this paper is to investigate $\lambda$-statistically quasi-Cauchy sequences. A real valued function $f$ defined on a subset $E$ of $\textbf{R}$, the set of real numbers, is called $\lambda$-statistically ward continuous on $E$ if it preserves $\lambda$-statistically quasi-Cauchy sequences of points in $E$. It turns out that uniform continuity coincides with $\lambda$-statistically ward continuity on $\lambda$-statistically ward compact subsets., Comment: 11 pages. arXiv admin note: substantial text overlap with arXiv:1205.3674, arXiv:1102.1531

A real function $f$ is ward continuous if $f$ preserves quasi-Cauchyness, i.e. $(f(x_{n}))$ is a quasi-Cauchy sequence whenever $(x_{n})$ is quasi-Cauchy; and a subset $E$ of $\textbf{R}$ is quasi-Cauchy compact if any sequence $\textbf{x}=(x_{n})$ of points in $E$ has a quasi-Cauchy subsequence where $\textbf{R}$ is the set of real numbers. These known results suggest to us introducing a concept of upward (respectively, downward) half quasi-Cauchy continuity in the sense that a function $f$ is upward (respectively, downward) half quasi-Cauchy continuous if it preserves upward (respectively, downward) half quasi-Cauchy sequences, and a concept of upward (respectively, downward) half quasi-Cauchy compactness in the sense that a subset $E$ of $\textbf{R}$ is upward (respectively, downward) half quasi-Cauchy compact if any sequence of points in $E$ has an upward (respectively, downward) half quasi-Cauchy subsequence. We investigate upward(respectively, downward) half quasi-Cauchy continuity and upward (respectively, downward) half quasi-Cauchy compactness, and prove related theorems., Comment: 24 pages. arXiv admin note: substantial text overlap with arXiv:1103.1230, arXiv:1102.1531, arXiv:1204.2445

In this paper we generalize the concept of a quasi-Cauchy sequence to a concept of a $p$-quasi-Cauchy sequence for any fixed positive integer $p$. For $p=1$ we obtain some earlier existing results as a special case. We obtain some interesting theorems related to $p$-quasi-Cauchy continuity, $G$-sequential continuity, slowly oscillating continuity, and uniform continuity. It turns out that a function $f$ defined on an interval is uniformly continuous if and only if there exists a positive integer $p$ such that $f$ preserves $p$-quasi-Cauchy sequences where a sequence $(x_{n})$ is called $p$-quasi-Cauchy if $(x_{n+p}-x_{n})_{n=1}^{\infty}$ is a null sequence., Comment: 15 pages. arXiv admin note: substantial text overlap with arXiv:1005.4940, arXiv:1203.2003, arXiv:1103.1230, arXiv:1102.1531

An ideal $I$ is a family of subsets of positive integers $\textbf{N}$ which is closed under taking finite unions and subsets of its elements. A sequence $(x_n)$ of real numbers is said to be $I$-convergent to a real number $L$, if for each \;$ \varepsilon> 0$ the set $\{n:|x_{n}-L|\geq \varepsilon\}$ belongs to $I$. We introduce $I$-ward compactness of a subset of $\textbf{R}$, the set of real numbers, and $I$-ward continuity of a real function in the senses that a subset $E$ of $\textbf{R}$ is $I$-ward compact if any sequence $(x_{n})$ of points in $E$ has an $I$-quasi-Cauchy subsequence, and a real function is $I$-ward continuous if it preserves $I$-quasi-Cauchy sequences where a sequence $(x_{n})$ is called to be $I$-quasi-Cauchy when $(\Delta x_{n})$ is $I$-convergent to 0. We obtain results related to $I$-ward continuity, $I$-ward compactness, ward continuity, ward compactness, ordinary compactness, ordinary continuity, $\delta$-ward continuity, and slowly oscillating continuity., Comment: 16 pages. arXiv admin note: text overlap with arXiv:1005.4940

A topological group $X$ is called connected if the only subsets which are both open and closed are the whole space $X$ and the null set $\emptyset$. A subset of a topological group is connected if the subspace is connected. We say that a subset $A$ of $X$ is $G$-sequentially connected if the only subsets of $A$ which are both $G$-sequentially open and $G$-sequentially closed, with respect to the relative $G$-sequentially open and $G$-sequentially closed subsets of $A$, are open and closed subsets of $A$ are $A$ and the null set, $\emptyset$. We investigate the impact of changing the definition of convergence of sequences on the structure of sequential connectedness of subsets of $X$ via sequential closure of sets in the sense of $G$-sequential closure. Sequential connectedness for topological groups is a special case of this generalization when G = lim., Comment: This paper has been withdrawn by the author since it will be published in Applied Mathematics Letter which is not open acces journal

In this paper we call a real-valued function $N_{\theta}$-ward continuous if it preserves $N_{\theta}$-quasi-Cauchy sequences where a sequence $\boldsymbol{\alpha}=(\alpha_{k})$ is defined to be $N_{\theta}$-quasi-Cauchy when the sequence $\Delta \boldsymbol{\alpha}$ is in $N^{0}_{\theta}$. We prove not only inclusion and compactness type theorems, but also continuity type theorems., Comment: 20 pages

A sequence $(x_{n})$ of points in a topological group is called $\Delta$-quasi-slowly oscillating if $(\Delta x_{n})$ is quasi-slowly oscillating, and is called quasi-slowly oscillating if $(\Delta x_{n})$ is slowly oscillating. A function $f$ defined on a subset of a topological group is quasi-slowly (respectively, $\Delta$-quasi-slowly) oscillating continuous if it preserves quasi-slowly (respectively, $\Delta$-quasi-slowly) oscillating sequences, i.e. $(f(x_{n}))$ is quasi-slowly (respectively, $\Delta$-quasi-slowly) oscillating whenever $(x_{n})$ is. We study these kinds of continuities, and investigate relations with statistical continuity, lacunary statistical continuity, and some other types of continuities in metrizable topological groups., Comment: This paper has been withdrawn by the author due to the publication in the journal, Computer and Mathematics with Applications

A sequence $\textbf{p}=(p_{n})$ of real numbers is called Abel convergent to $\ell$ if the series $\Sigma_{k=0}^{\infty}p_{k}x^{k}$ is convergent for $0\leq x<1$ and \[\lim_{x \to 1^{-}}(1-x) \sum_{k=0}^{\infty}p_{k}x^{k}=\ell.\] We introduce a concept of Abel continuity in the sense that a function $f$ defined on a subset of $\Re$, the set of real numbers, is Abel continuous if it preserves Abel convergent sequences, i.e. $(f(p_{n}))$ is an Abel convergent sequence whenever $(p_{n})$ is. A new type of compactness, namely Abel sequential compactness is also introduced, and interesting theorems related to this kind of compactnes, Abel continuity, statistical continuity, lacunary statistical continuity, ordinary continuity, and uniform continuity are obtained., Comment: 12 pages

This paper is withdrawn because the results in the paper are included in a paper to be published in Mathematical and Computer Modelling., Comment: This paper is withdrawn because the results in the paper are included in a paper to be published in Mathematical and Computer Modelling

This paper is withdrawn because the results in the paper are included in a paper to be published in Mathematical and Computer Modelling., Comment: This paper is withdrawn because the results in the paper are included in a paper to be published in Mathematical and Computer Modelling

Scalarization method is an important tool in the study of vector optimization as corresponding solutions of vector optimization problems can be found by solving scalar optimization problems. This is applied by Du (2010) [A note on cone metric fixed point theory and its equivalence, Nonlinear Analysis 72 2259-2261] to investigate the equivalence of vectorial versions of fixed point theorems of contractive mappings in generalized cone metric spaces and scalar versions of fixed point theorems in general metric spaces in usual sense. In this paper we find out that the topology induced by topological vector space valued cone metric coincides with the topology induced by the metric obtained via a nonlinear scalarization function, i.e any topological vector space valued cone metric space is metrizable., Comment: This paper has been withdrawn by the authors. We withdraw our paper due to the acceptance in the journal "Applied Mathematics Letters" with a different title. r has been withdrawn by the authors due to a

A function $f$ on a topological space is sequentially continuous at a point $u$ if, given a sequence $(x_{n})$, $\lim x_{n}=u$ implies that $\lim f(x_{n})=f(u)$. This definition was modified by Connor and Grosse-Erdmann for real functions by replacing $lim$ with an arbitrary linear functional $G$ defined on a linear subspace of the vector space of all real sequences. In this paper, we extend this definition to a topological group $X$ by replacing $G$ a linear functional with an arbitrary additive function defined on a subgroup of the group of all $X$-valued sequences and not only give new theorems in this generalized setting but also obtain theorems which are not appeared even for real functions so far., Comment: This paper has been withdrawn by the author. I withdraw my paper due to the acceptance in the journal "Computers & Mathematics with Applications"

This paper is withdrawn because the results in the paper are included in a paper to be published in Mathematical and Computer Modelling., Comment: presented at MathFest August 5-7, 2010, Pittsburgh, USA This paper is withdrawn because the results in the paper are included in a paper to be published in another journal

Recently, a concept of forward continuity and a concept of forward compactness are introduced in the senses that a function $f$ is forward continuous if $\lim_{n\to\infty} \Delta f(x_{n})=0$ whenever $\lim_{n\to\infty} \Delta x_{n}=0$,\; and a subset $E$ of $\textbf{R}$ is forward compact if any sequence $\textbf{x}=(x_{n})$ of points in $E$ has a subsequence $\textbf{z}=(z_{k})=(x_{n_{k}})$ of the sequence $\textbf{x}$ such that $\lim_{k\to \infty} \Delta z_{k}=0$ where $\Delta z_{k}=z_{k+1}-z_{k}$. These concepts suggest us to introduce a concept of second forward continuity in the sense that a function $f$ is second forward continuous if $\lim_{n\to\infty}\Delta^{2}f(x_{n})=0$ whenever $\lim_{n\to\infty}\Delta^{2}x_{n}=0$, and a subset $E$ of $\textbf{R}$ is second forward compact if whenever $\textbf{x}=(x_{n})$ is a sequence of points in $E$ there is a subsequence $\textbf{z}=(z_{k})=(x_{n_{k}})$ of $\textbf{x}$ with $\lim_{k\to \infty} \Delta^{2}z_{k}=0$ where $\Delta^{2} y_{n}=y_{n+2}-2y_{n+1}+y_{n}$. We investigate the impact of changing the definition of convergence of sequences on the structure of forward continuity in the sense of second forward continuity, and compactness of sets in the sense of second forward compactness, and prove related theorems., Comment: I withdraw my paper due to the acceptance in the journal "Mathematical and Computer Modelling"

In this paper, we study main properties of cone normed spaces, and prove some theorems of weighted means in cone normed spaces., Comment: We withdraw our paper due to the acceptance in the journal "Mathematical and Computer Modelling"